Blender Python: How to activate the Auto Refresh in a imported sequence on Image Editor?
Digging for classes and properties.

If you right click and view source on checkbox you see,

def draw(self, context):
    layout = self.layout

    sima = context.space_data
    iuser = sima.image_user

    layout.template_image(sima, "image", iuser, multiview=True)


which gives a clue that it belongs to the image editor space sima and there is a class  ImageUser instance sima.image_user... 

Ok off to py console, making sure an image editor is open in same screen, in case below its in area 1

&gt;&gt;&gt; for i, a in enumerate(C.screen.areas):
...     i, a.type
...     
(0, 'INFO')
(1, 'IMAGE_EDITOR')
(2, 'CONSOLE')
(3, 'VIEW_3D')
(4, 'TEXT_EDITOR')

&gt;&gt;&gt; sp = C.screen.areas[1].spaces.active
&gt;&gt;&gt; sp.image_user.use_auto_refresh
False


Hence in your operator, as long as it is invoked from the image editor you can set with

space = context.space_data
space.image_user.use_auto_refresh = True


This is an example of

SpaceImageEditor.image_user


See the reference section of manual https://docs.blender.org/api/2.79/bpy.types.ImageUser.html?highlight=use_auto_refresh#bpy.types.ImageUser.use_auto_refresh for other possible references.

BackgroundImage.image_user
EnvironmentMapTexture.image_user
Image.filepath_from_user
ImageTexture.image_user
Object.image_user
ShaderNodeTexEnvironment.image_user
ShaderNodeTexImage.image_user    
TextureNodeImage.image_user
UILayout.template_image
UILayout.template_image_layers
VoxelDataTexture.image_user